Introducing Monotone Enriched Nonexpansive Mappings for Fixed Point Approximation in Ordered CAT(0) Spaces

Abstract

The aim of this paper is twofold: introducing the concept of monotone enriched nonexpansive mappings and a faster iterative process. Our examples illustrate the novelty of our newly introduced concepts. We investigate the iterative estimation of fixed points for such mappings for the first time within an ordered CAT(0) space. It is done by proving some strong and $\Delta$-convergence theorems. Additionally, numerical experiments are included to demonstrate the validity of our theoretical results and to establish the superiority of convergence behavior of our iterative process. As an application, we use our newly introduced concepts to find the solution of an integral equation. The outcomes of our study expand upon and enhance certain established findings in the current body of literature.

1. Introduction

Fixed-point theory plays a fundamental role in various analytical and computational methods with applications in optimization, game theory, economics, and differential equations [1]. A fixed point may not always exist, and even when it does, computing it can be challenging. Approximating fixed points relies on three key factors: iterative algorithms, the underlying space and the mappings used. Iterative processes are crucial in fixed-point theory, both for proving existence and for approximation. Classical examples, such as the Babylonian Algorithm [2,3], an ancient method for approximating square roots, implicitly relies on iterative fixed-point techniques. This algorithm is surprisingly effective and efficient. It shows the ingenuity of ancient mathematicians in developing practical methods for solving complex mathematical problems. Although developed thousands of years ago, the old Babylonian algorithm still has relevance today and can be used as an alternative approach to modern computational methods in certain situations. This historical approach aligns with modern iterative methods, such as the variational iteration method and other contemporary numerical schemes used in nonlinear analysis. The connection between these methods underscores the long-standing significance of fixed-point theory in both theoretical and practical problem-solving contexts.

Extending results from linear to nonlinear spaces is a key mathematical challenge. Many problems in metric spaces lack natural linear and convex structures, so it is natural to focus on CAT(0) spaces, a special class of metric spaces that includes Hilbert and some Banach spaces.

Mappings determine the behavior of fixed-point approximation methods. Monotone nonexpansive mappings, which combine monotonicity and nonexpansiveness, play a key role in convex analysis and optimization. The class of enriched nonexpansive mappings (E-NEM) was introduced by Berinde [4] as a generalization of the class of nonexpansive mappings (NEM). Due to the importance of these mappings, many research articles were established using these mappings. (see [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] for more details).

Consider a real Hilbert space $\mathcal{H}$ with a norm $\parallel ·\parallel$ induced by an inner product $〈·,·〉$. Let $\mathfrak{U}\ne \varnothing$ be a closed and convex subset of $\mathcal{H}$. A mapping $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ is termed:

(i)

a nonexpansive for all $\hslash ,\mathsf{\ell }\in \mathfrak{U},$ we have

$$∥\mathcal{J} \hslash -\mathcal{J} \mathsf{\ell }∥\le ∥\hslash -\mathsf{\ell }∥;$$

(1)

(ii)

an enriched nonexpansive if ∃$\mathfrak{w}\in [0,+\infty )$∀$\hslash ,\mathsf{\ell }\in \mathfrak{U},$ we have

$$∥\mathfrak{w}(\hslash -\mathsf{\ell })+\mathcal{J} \hslash -\mathcal{J} \mathsf{\ell }∥\le (\mathfrak{w}+1)∥\hslash -\mathsf{\ell }∥.$$

(2)

On the other hand, motivated by the work of [26,27], Abbas et.al [28] have introduced in 2022 the notion of E-NEM in CAT(0) spaces. To approximate the fixed point (FP) of an E-NEM, a new iteration called MRN-iteration was established. Furthermore, the numerical experiments conducted in [28] led to the conclusion that utilizing the MRN-iteration is more advantageous compared to other iterations ([29,30,31]) proposed in the literature for the category of E-NEM.

Bachar and Khamsi [32] investigated the presence of FP for monotone NEM operating on partially ordered Banach spaces (see [33,34,35] and references mentioned therein). Due to the significance of iterative procedures, numerous novel iterative sequences have been developed in recent years. Researchers are primarily concerned with deriving iterative sequences that converge more rapidly than the current ones. The aim of this paper is twofold: first to define monotone E-NEM and show the existence and iterative approximation of their fixed points; the second is to demonstrate that the new iteration defined in this paper converges faster than MRN-iteration under certain conditions and many other iterations in the literature.

2. Preliminaries

Consider a metric space $(\mathcal{S},\vartheta )$. A geodesic path from ℏ to ℓ is defined as a mapping c from a closed interval $[0,l]\subset \mathbb{R}$ to $\mathcal{S}$ st $c\left(0\right)=\hslash$, $c\left(l\right)=\mathsf{\ell }$, and $\vartheta \left(c\right(\mathcal{J}),c{\left(\mathcal{J}\right)}^{\prime })=|\mathcal{J}-{\mathcal{J}}^{\prime }|$ for all $\mathcal{J},{\mathcal{J}}^{\prime }\in [0,1]$. Specifically, c acts as an isometry, and $\vartheta (\hslash ,\mathsf{\ell })=l$. The range of c is referred to as a geodesic (or metric) segment connecting ℏ and ℓ. The space $(\mathcal{S},\vartheta )$ is termed a geodesic space if every pair of points in $\mathcal{S}$ is connected by a geodesic. Moreover, $\mathcal{S}$ is uniquely geodesic if there exists only one geodesic linking ℏ and ℓ for each $\hslash ,\mathsf{\ell }\in \mathcal{S}$, denoted as $[\hslash ,\mathsf{\ell }]$, representing the segment from ℏ to ℓ. A geodesic triangle $\Delta (\hslash ,\mathsf{\ell },\mathfrak{q})$ in the geodesic metric space $(\mathcal{S},\vartheta )$ comprises three points $\hslash ,\mathsf{\ell },\mathfrak{q}$ in $\mathcal{S}$ (the vertices of $\Delta$) and geodesic segments between each pair of vertices (the edges of $\Delta$). A comparison triangle for the geodesic triangle $\Delta (\hslash ,\mathsf{\ell },\mathfrak{q})$ in $\mathcal{S}$ can be represented as a triangle $\Delta (\hslash ,\mathsf{\ell },\mathfrak{q}):=\overline{\Delta }(\overline{\hslash },\overline{\mathsf{\ell }},\overline{\mathfrak{q}})$ in the Euclidean plane ${\mathbb{R}}^2$, where ${\vartheta }_{{\mathbb{R}}^2}(\overline{\hslash },\overline{\mathsf{\ell }})=\vartheta (\hslash ,\mathsf{\ell })$, ${\vartheta }_{{\mathbb{R}}^2}(\overline{\mathsf{\ell }},\overline{\mathfrak{q}})=\vartheta (\mathsf{\ell },\mathfrak{q})$ and ${\vartheta }_{{\mathbb{R}}^2}(\overline{\mathfrak{q}},\overline{\hslash })=\vartheta (\mathfrak{q},\hslash )$. A geodesic triangle $\Delta (\hslash ,\mathsf{\ell },\mathfrak{q})$ in $(\mathcal{S},\vartheta )$ is considered to satisfy the CAT(0) inequality if, for any $u,v\in \Delta (\hslash ,\mathsf{\ell },\mathfrak{q})$ and their corresponding comparison points $\overline{u},\overline{v}\in \overline{\Delta }(\overline{\hslash },\overline{\mathsf{\ell }},\overline{\mathfrak{q}})$, it holds that $\vartheta (u,v)\le {\vartheta }_{{\mathbb{R}}^2}(\overline{u},\overline{v})$. A geodesic space $\mathcal{S}$ is classified as a CAT(0) space if the CAT(0) inequality is satisfied by all its geodesic triangles. To explore more about equivalent definitions and key properties of CAT(0) spaces, one may consult reputable texts like [36,37]. One widely acknowledged fact is that every CAT(0) space has a unique geodesic nature.

Let $\left\{{\hslash }_{\breve{e}}\right\}$ be a bounded sequence in a CAT(0) space. For $\hslash \in \mathcal{S}$, we set $r(\hslash ,\left\{{\hslash }_{\breve{e}}\right\})={lim\; sup}_{\breve{e}\to \infty }\vartheta (\hslash ,{\hslash }_{\breve{e}}).$ The asymptotic radius $R\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$ of $\left\{{\hslash }_{\breve{e}}\right\}$ is given by

$$R\left(\left\{{\hslash }_{\breve{e}}\right\}\right)=inf\left\{r\left(\hslash ,\left\{{\hslash }_{\breve{e}}\right\}\right):\hslash \in \mathcal{S}\right\},$$

and the asymptotic center $A\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$ of $\left\{{\hslash }_{\breve{e}}\right\}$ is the set $A\left(\left\{{\hslash }_{\breve{e}}\right\}\right)=\{\hslash \in \mathcal{S}:r\left(\hslash ,\left\{{\hslash }_{\breve{e}}\right\}\right)=R(\left\{{\hslash }_{\breve{e}}\right\}\left)\right\}.$ It is known (see, e.g., [38], Proposition 7) that in a complete CAT (0) space, $A\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$ consists of exactly one point.

In order to establish the main findings of this paper, it is essential to introduce certain definitions and auxiliary results that are compiled in the lemmas.

Lemma 1

([39]). Let $(\mathcal{S},\vartheta )$ be a CAT(0) space. For $\hslash ,\mathsf{\ell }\in \mathcal{S}$ and $\mathcal{J}\in [0,1],$ ∃ a unique point $\mathfrak{q}\in [\hslash ,\mathsf{\ell }]$ st

$$\vartheta (\hslash ,\mathfrak{q})=\mathcal{J} \vartheta (\hslash ,\mathsf{\ell })\mathit{and}\vartheta (\mathsf{\ell },\mathfrak{q})=(1-\mathcal{J})\vartheta (\hslash ,\mathsf{\ell }).$$

(3)

For convenience, from now on we will use the notation $(1-\mathcal{J})\hslash \oplus \mathcal{J} \mathsf{\ell }$ for the unique point $\mathfrak{q}$ satisfying (3).

Lemma 2

([39]). Let $(\mathcal{S},\vartheta )$ be a CAT(0) space.

1.

For $\hslash ,\mathsf{\ell },\mathfrak{q}\in \mathcal{S}$ and $\mathcal{J}\in [0,1]$, we have

$$\vartheta \left(\right(1-\mathcal{J})\hslash \oplus \mathcal{J} \mathsf{\ell },\mathfrak{q})\le (1-\mathcal{J})\vartheta (\hslash ,\mathfrak{q})+\mathcal{J} \vartheta (\mathsf{\ell },\mathfrak{q}).$$

2.

For $\hslash ,\mathsf{\ell },\mathfrak{q}\in \mathcal{S}$ and $\mathcal{J}\in [0,1]$, we have

$${\vartheta }^2((1-\mathcal{J})\hslash \oplus \mathcal{J} \mathsf{\ell },\mathfrak{q})\le (1-\mathcal{J}){\vartheta }^2(\hslash ,\mathfrak{q})+\mathcal{J}{\vartheta }^2(\mathsf{\ell },\mathfrak{q})-\mathcal{J}(1-\mathcal{J}){\vartheta }^2(\hslash ,\mathsf{\ell }).$$

Definition 1

([40]). A Banach space is said to satisfy Opial’s condition if the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ in $\mathcal{S}$ with ${\hslash }_{\breve{e}}\rightharpoonup \hslash$, it implies that ${lim\; sup}_{\breve{e}\to \infty }∥{\hslash }_{\breve{e}}-\hslash ∥<{lim\; sup}_{\breve{e}\to \infty }∥{\hslash }_{\breve{e}}-\mathsf{\ell }∥$ for all $\mathsf{\ell }\in \mathcal{S}$ with $\mathsf{\ell }\ne \hslash$.

Definition 2

([41]). A sequence $\left\{{\hslash }_{\breve{e}}\right\}$ in $\mathcal{S}$ is said to Δ-converge to $\hslash \in \mathcal{S}$ if ℏ is the unique asymptotic center of $\left\{u_{\breve{e}}\right\}$ for every subsequence $\left\{u_{\breve{e}}\right\}$ of $\left\{{\hslash }_{\breve{e}}\right\}$. In this case we write Δ-${lim}_{\breve{e}}{\hslash }_{\breve{e}}=\hslash$ and call ℏ the Δ-limit of $\left\{{\hslash }_{\breve{e}}\right\}$.

Based on the definition of $\Delta$-convergence, It is evident that each CAT(0) space upholds Opial’s property. This is because if $\left\{{\hslash }_{\breve{e}}\right\}$ is a sequence in $\mathcal{S}$ with $\Delta$-${lim}_{\breve{e}}{\hslash }_{\breve{e}}=\hslash$, then we observe that ${lim\; sup}_{\breve{e}\to \infty }\vartheta ({\hslash }_{\breve{e}},\hslash )<{lim\; sup}_{\breve{e}\to \infty }\vartheta ({\hslash }_{\breve{e}},\mathsf{\ell })$ for all $\mathsf{\ell }\in \mathcal{S}$ where $\mathsf{\ell }\ne \hslash$.

Lemma 3.

(i)

Every bounded sequence in a complete CAT(0) space $\mathcal{S}$ has a Δ-convergent subsequence (see [42]).

(ii)

If $\mathfrak{U}$ is a closed convex subset of $\mathcal{S}$ and if $\left\{{\hslash }_{\breve{e}}\right\}$ is a bounded sequence in $\mathfrak{U}$, then the asymptotic center of $\left\{{\hslash }_{\breve{e}}\right\}$ is in $\mathfrak{U}$ (see [43]).

The following lemma is a consequence of ([44], Lemma 2.9) which is used to prove our main result.

Lemma 4

([44]). Let $(\mathcal{S},\vartheta )$ be a complete CAT(0) space and $\hslash \in \mathcal{S}$. Suppose $\left\{{\mathcal{J}}_{\breve{e}}\right\}$ is a sequence in $\left[\mathfrak{w},c\right]$ for some $\mathfrak{w},c\in (0,1)$, and $\left\{u_{\breve{e}}\right\}$, $\left\{v_{\breve{e}}\right\}$ are sequences in $\mathcal{S}$ st ${lim\; sup}_{\breve{e}\to \infty }\vartheta (u_{\breve{e}},\hslash )\le r$, ${lim\; sup}_{\breve{e}\to \infty }\vartheta (v_{\breve{e}},\hslash )\le r$ and ${lim}_{\breve{e}\to \infty }\vartheta ({\mathcal{J}}_{\breve{e}}{v}_{\breve{e}}\oplus \left(1-{\mathcal{J}}_{\breve{e}}\right)u_{\breve{e}},\hslash )=r$ holds for a certain $r\ge 0$, then

${lim}_{\breve{e}\to \infty }\vartheta \left(u_{\breve{e}},v_{\breve{e}}\right)=0$.

We need the following lemma from [28].

Lemma 5

([28]). Let $\mathfrak{U}$ be a convex subset of a CAT(0) space $\mathcal{S}$ and $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}.$ Define the mapping ${\mathcal{J}}_{\mathring{\varrho }}:\mathfrak{U}\to \mathfrak{U}$ by

$${\mathcal{J}}_{\mathring{\varrho }}\hslash =(1-\mathring{\varrho })\hslash \oplus \mathring{\varrho }\mathcal{J} \hslash ,\forall \hslash \in \mathfrak{U}.$$

(4)

Then for any $\mathring{\varrho }\in (0,1]$,

$$Fix\left(\mathcal{J}\right)=\{\hslash \in \mathfrak{U}:\mathcal{J} \hslash =\hslash \}=\{\hslash \in \mathfrak{U}:{\mathcal{J}}_{\mathring{\varrho }}\hslash =\hslash \}=Fix\left({\mathcal{J}}_{\mathring{\varrho }}\right).$$

(5)

Consider $\mathcal{S}$ as a complete CAT(0) space with the partial order denoted by ‘⪯’. Additionally, we make the assumption that the order intervals are both closed and convex. An order interval refers to any of the subsets

$$[\tilde{o},\to )=\{\hslash \in \mathcal{S};\tilde{o}⪯\hslash \}\mathrm{or}(\leftarrow ,\tilde{o}]=\{\hslash \in \mathcal{S}:\hslash ⪯\tilde{o}\}$$

for any $\tilde{o}\in \mathcal{S}$. So, an order interval $[\hslash ,\mathsf{\ell }]$ for all $\hslash ,\mathsf{\ell }\in \mathcal{S}$, is given by

$$[\hslash ,\mathsf{\ell }]=\{\mathfrak{q}\in \mathcal{S}:\hslash ⪯\mathfrak{q}⪯\mathsf{\ell }\}.$$

Clearly, the order interval $[\hslash ,\mathsf{\ell }]$ exhibits properties of being closed and convex.

3. Monotone Enriched Nonexpansive Mappings in Ordered CAT(0) Spaces

Next we give the definition of enriched monotone mappings.

Definition 3.

Let $\mathfrak{U}\ne \varnothing$ be a subset of an ordered CAT(0) space $\mathcal{S}.$ A mapping $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ is classified as EMM or EOP provided that there exists $\mathring{\varrho }\in (0,1]$ st

$$\hslash ⪯\mathsf{\ell }\Rightarrow (1-\mathring{\varrho })\hslash \oplus \mathring{\varrho }\mathcal{J} \hslash ⪯(1-\mathring{\varrho })\mathsf{\ell }\oplus \mathring{\varrho }\mathcal{J} \mathsf{\ell },$$

(6)

for all $\hslash ,\mathsf{\ell }\in \mathfrak{U}.$

In view of (4), the condition (6) reduced to

$$\hslash ⪯\mathsf{\ell }\Rightarrow {\mathcal{J}}_{\mathring{\varrho }}\hslash ⪯{\mathcal{J}}_{\mathring{\varrho }}\mathsf{\ell }.$$

(7)

Next, we present the definition of a monotonic E-NEM.

Definition 4.

Let $\mathfrak{U}$ be a nonempty subset of an ordered CAT(0) space $\mathcal{S}.$ A mapping $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ is said to be monotone $\mathring{\varrho }$-enriched nonexpansive mapping (ENEM) if $\mathcal{J}$ is enriched monotone (EM) and for all $\hslash ,\mathsf{\ell }\in \mathfrak{U}$ with $\hslash ⪯\mathsf{\ell },$ we have

$$\vartheta ((1-\mathring{\varrho })\hslash \oplus \mathring{\varrho }\mathcal{J} \hslash ,(1-\mathring{\varrho })\mathsf{\ell }\oplus \mathring{\varrho }\mathcal{J} \mathsf{\ell })\le \vartheta (\hslash ,\mathsf{\ell }).$$

(8)

To highlight the constants involved in (6) and (8), we call $\mathcal{J}$ a $\mathring{\varrho }$-EM and $\mathring{\varrho }$-MENE, respectively.

Example 1.

Any $\mathring{\varrho }$-ENEM $\mathcal{J}$ (defined in [28]) is a $\mathring{\varrho }$-MENE, that is $\mathcal{J}$ satisfies (8) but the converse is not true (see, Example 3).

There exists a class of mappings that only satisfies the condition (8) but not (6). The following example, illustrate this fact.

Example 2.

Let $\mathcal{S}=[0,+\infty )$ be a CMS with the metric

$$\vartheta (\hslash ,\mathsf{\ell })=|\hslash -\mathsf{\ell }|,\hslash ,\mathsf{\ell }\in \mathcal{S},$$

Given $\mathfrak{U}=[0,1]$, we now examine the ordering relation ⪯ defined as $\hslash ⪯\mathsf{\ell }\iff \hslash \le \mathsf{\ell }$:

$$\left\{\begin{array}{cc}\hslash \le \mathsf{\ell }\hfill & \mathit{if}\hslash ,\mathsf{\ell }\in \{0,{\displaystyle \frac{1}{2}},1\}\mathit{or}\hfill \\ \hslash \le \mathsf{\ell }\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Define a mapping $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ as follows

$$\mathcal{J}(\hslash )=\left\{\begin{array}{cc}0\hfill & \hslash \in \{{\displaystyle \frac{1}{2}},1\}\hfill \\ 1\hfill & \hslash =0\hfill \\ \frac{1}{3}\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Clearly, $\mathcal{J}$ is not continuous, so it is not a nonexpansive mapping.

Now we show that under the above ordering $\mathcal{J}$ satisfies the inequality (8) for the value of $\mathring{\varrho }=1/2.$ We verify it as follows:

• If $\hslash =0$ and $\mathsf{\ell }=\frac{1}{2}$ and $\hslash \le \mathsf{\ell }$, we have

$$\begin{array}{cc}\hfill \vartheta \left({\displaystyle \frac{1}{2}}\hslash +{\displaystyle \frac{1}{2}}\mathcal{J} \hslash ,{\displaystyle \frac{1}{2}}\mathsf{\ell }+{\displaystyle \frac{1}{2}}\mathcal{J} \mathsf{\ell }\right)& =\left|\left({\displaystyle \frac{1}{2}}·0+{\displaystyle \frac{1}{2}}·1\right)-\left({\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}·0\right)\right|={\displaystyle \frac{1}{4}}\hfill \\ \hfill & <{\displaystyle \frac{1}{2}}=\vartheta (\hslash ,\mathsf{\ell }).\hfill \end{array}$$

• If $\hslash =0$ and $\mathsf{\ell }=1$ and $\hslash \le \mathsf{\ell }$, we have

$$\begin{array}{cc}\hfill \vartheta \left({\displaystyle \frac{1}{2}}\hslash +{\displaystyle \frac{1}{2}}\mathcal{J} \hslash ,{\displaystyle \frac{1}{2}}\mathsf{\ell }+{\displaystyle \frac{1}{2}}\mathcal{J} \mathsf{\ell }\right)& =\left|\left({\displaystyle \frac{1}{2}}·0+{\displaystyle \frac{1}{2}}·1\right)-\left({\displaystyle \frac{1}{2}}·1+{\displaystyle \frac{1}{2}}·0\right)\right|=0\hfill \\ \hfill & <1=\vartheta (\hslash ,\mathsf{\ell }).\hfill \end{array}$$

• If $\hslash =\frac{1}{2}$ and $\mathsf{\ell }=1$ and $\hslash \le \mathsf{\ell }$, we have

$$\begin{array}{cc}\hfill \vartheta \left({\displaystyle \frac{1}{2}}\hslash +{\displaystyle \frac{1}{2}}\mathcal{J} \hslash ,{\displaystyle \frac{1}{2}}\mathsf{\ell }+{\displaystyle \frac{1}{2}}\mathcal{J} \mathsf{\ell }\right)& =\left|\left({\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}·0\right)-\left({\displaystyle \frac{1}{2}}·1+{\displaystyle \frac{1}{2}}·0\right)\right|={\displaystyle \frac{1}{4}}\hfill \\ \hfill & <{\displaystyle \frac{1}{2}}=\vartheta (\hslash ,\mathsf{\ell }).\hfill \end{array}$$

• If $\hslash ,\mathsf{\ell }\in \mathfrak{U}\setminus \{0,\frac{1}{2},1\}$ and $\hslash \le \mathsf{\ell }$, we have

$$\begin{array}{cc}\hfill \vartheta \left({\displaystyle \frac{1}{2}}\hslash +{\displaystyle \frac{1}{2}}\mathcal{J} \hslash ,{\displaystyle \frac{1}{2}}\mathsf{\ell }+{\displaystyle \frac{1}{2}}\mathcal{J} \mathsf{\ell }\right)& =\left|\left({\displaystyle \frac{1}{2}}\hslash +{\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{3}}\right)-\left({\displaystyle \frac{1}{2}}\mathsf{\ell }+{\displaystyle \frac{1}{2}}·{\displaystyle \frac{1}{3}}\right)\right|={\displaystyle \frac{1}{2}}\left|\hslash -\mathsf{\ell }\right|\hfill \\ \hfill & <\left|\hslash -\mathsf{\ell }\right|=\vartheta (\hslash ,\mathsf{\ell }).\hfill \end{array}$$

In all the above cases, inequality (8) holds.

On the other hand, $\mathcal{J}$ is not a $\mathring{\varrho }$-EM for any $\mathring{\varrho }\in (0,1].$ Indeed,

$${\mathcal{J}}_{\mathring{\varrho }}(\hslash )=\left\{\begin{array}{cc}(1-\mathring{\varrho })\hslash \hfill & \hslash \in \{\frac{1}{2},1\}\hfill \\ \mathring{\varrho }\hfill & \hslash =0\hfill \\ \frac{1}{3}\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Notice that if $\hslash =0,\mathsf{\ell }=1$ with $\hslash \le \mathsf{\ell }$, we have

$${\mathcal{J}}_{\mathring{\varrho }}\hslash \nleqq {\mathcal{J}}_{\mathring{\varrho }}\mathsf{\ell }.$$

4. Some $\Delta$-Convergence and Strong Convergence Theorems

We now introduce our iterative sequence (IS) within the context of CAT(0) spaces. Assuming that $\mathfrak{U}\ne \varnothing$ is a convex subset (CS) of a CAT(0) space $\mathcal{S}$, then

$$\left\{\begin{array}{c}{\hslash }_1\in \mathfrak{U},\hfill \\ {\mathfrak{q}}_{\breve{\mathfrak{e}}}=\left(1-{\alpha }_{\breve{e}}\right){\hslash }_{\breve{e}}\oplus {\alpha }_{\breve{e}}{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},\hfill \\ {\mathsf{\ell }}_{\breve{e}}=\left(1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\right){\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\oplus {\mathfrak{Y}}_{\breve{\mathfrak{e}}}{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}},\hfill \\ {\hslash }_{\breve{e}+1}={\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}},\hfill \end{array}\right.$$

(9)

where $\left\{{\tilde{o}}_{\breve{e}}\right\}$ and $\left\{{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\right\}$ are sequences in $(0,1)$ st $0<\tilde{o}<{\beta }_{\breve{e}}.$

Next, we establish theorems on $\Delta$-convergence and strong convergence in CAT(0) spaces. Let’s start with the subsequent crucial lemma.

Lemma 6.

Let $\mathfrak{U}\ne \varnothing$ be a closed and CS of a complete ordered CAT(0) space $(\mathcal{S},⪯)$, and let $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ be a monotone $\mathring{\varrho }$-ENEM. Choose ${\hslash }_1\in \mathfrak{U}$ st ${\hslash }_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1$. If $\left\{{\hslash }_{\breve{e}}\right\}$ is given by (9), then we obtain the following:

(i)

${\hslash }_{\breve{e}}⪯{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}⪯{\mathsf{\ell }}_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}}={\hslash }_{\breve{e}+1}$,

(ii)

${\hslash }_{\breve{e}}⪯\mathsf{p}$, provided $\left\{{\hslash }_{\breve{e}}\right\}\Delta$-converges to a point $\mathsf{p}\in \mathfrak{U}$.

Proof. 

(i)

We establish the result through induction on $\breve{e}$.

If $c_1,c_2\in \mathfrak{U}$ st $c_1⪯c_2$, then the inequality $c_1⪯\alpha c_1\oplus (1-\alpha )c_2⪯c_2$ holds for $0\le \alpha \le 1$. We have assumed the convexity of order intervals, hence this is valid. Hence, it suffices to prove that ${\hslash }_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}$ for any $\breve{e}\ge 1$.

Having already assumed ${\hslash }_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1$, the inequality holds true for $\breve{e}=1$. Let’s assume ${\hslash }_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}$ for $\breve{e}\ge 2$.

Based on Equation (2), we find:

$$\begin{array}{cc}\hfill {\hslash }_{\breve{e}}& ⪯\left(1-{\alpha }_{\breve{e}}\right){\hslash }_{\breve{e}}\oplus {\alpha }_{\breve{e}}{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}={\mathfrak{q}}_{\breve{\mathfrak{e}}}\hfill \\ \hfill & ⪯\left(1-{\alpha }_{\breve{e}}\right){\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\oplus {\alpha }_{\breve{e}}{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}={\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}.\hfill \end{array}$$

(10)

Since $\mathcal{J}$ is monotone, we get ${\hslash }_{\breve{e}}⪯{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}.$ Once more, based on (2), we obtain:

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}& ⪯\left(1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}}{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\oplus {\mathfrak{Y}}_{\breve{\mathfrak{e}}}{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}={\mathsf{\ell }}_{\breve{e}}\right.\hfill \\ \hfill & ⪯\left(1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\right){\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}\oplus {\mathfrak{Y}}_{\breve{\mathfrak{e}}}{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}={\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}},\hfill \end{array}$$

(11)

this yields ${\hslash }_{\breve{e}}⪯{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}⪯{\mathsf{\ell }}_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}$. Utilizing the monotonicity property of $\mathcal{J}$, we can further establish ${\hslash }_{\breve{e}}⪯{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}⪯{\mathsf{\ell }}_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}}={\hslash }_{\breve{e}+1}$.

Considering ${\mathsf{\ell }}_{\breve{e}}⪯{\hslash }_{\breve{e}+1}$, and leveraging the monotonicity of $\mathcal{J}$, we deduce ${\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}+1}$, resulting in ${\hslash }_{\breve{e}+1}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}+1}$ since ${\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}}={\hslash }_{\breve{e}+1}$. Therefore, by induction, the inequality holds ∀$\breve{e}\ge 1$.

(ii)

Suppose $\mathsf{p}$ is a $\Delta$-limit of the sequence $\left\{{\hslash }_{\breve{e}}\right\}$. By the result in part (i), we know that ${\hslash }_{\breve{e}}⪯{\hslash }_{\breve{e}+1}$ for all $\breve{e}\ge 1$. Therefore, the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ is monotonically increasing, and the order interval $\left[{\hslash }_{\breve{c}},\to \right)$ is both closed and convex. Thus, it must be the case that $\mathsf{p}\in \left[{\hslash }_{\breve{c}},\to \right)$ for a specific $\breve{c}\in \mathbb{N}$. If $\mathsf{p}$ is not in the interval $\left[{\hslash }_{\breve{c}},\to \right)$, then the AC of the subsequence $\left\{{\hslash }_r\right\}$, obtained by excluding the initial $\breve{c}-1$ terms from the sequence $\left\{{\hslash }_{\breve{e}}\right\}$, cannot be $\mathsf{p}$. This refutes the assumption that $\mathsf{p}$ is a $\Delta$-limit of the sequence $\left\{{\hslash }_{\breve{e}}\right\}$, thus concluding the proof of part (ii).

□

Lemma 7.

Let $\mathfrak{U}\ne \varnothing$ be a closed and CS of a complete ordered CAT(0) space $(\mathcal{S},⪯)$, and let $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ be a monotone $\mathring{\varrho }$-ENEM. Choose ${\hslash }_1\in \mathfrak{U}$ st ${\hslash }_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1$. If $\left\{{\hslash }_{\breve{e}}\right\}$ is given by (9), and $Fix\left(\mathcal{J}\right)\ne \varnothing$ withp$\in Fix\left(\mathcal{J}\right)$ st $\mathsf{p}⪯{\hslash }_1$, then the following results are valid:

(i)

${lim}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},\mathsf{p}\right)$ exists,

(ii)

${lim}_{\breve{e}\to \infty }\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)=0$.

Proof. 

Given the initial condition $\mathsf{p}⪯{\hslash }_1$, utilizing part (i) of Lemma 6 for $\breve{e}=1$ leads to the following sequence of relations: ${\hslash }_1⪯{\mathfrak{q}}_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1⪯{\mathsf{\ell }}_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_1⪯{\hslash }_2$. It is clear that $\mathsf{p}⪯{\hslash }_1⪯{\hslash }_2$, resulting in $\mathsf{p}⪯{\hslash }_2$. By utilizing mathematical induction with respect to $\breve{e}$, it can be easily established that $\mathsf{p}⪯{\hslash }_{\breve{e}}$ for all $\breve{e}\ge 1$. Furthermore, employing Lemma 6 (i), we obtain:

$$\mathsf{p}⪯{\hslash }_{\breve{e}}⪯{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}⪯{\mathsf{\ell }}_{\breve{e}}⪯{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}⪯{\hslash }_{\breve{e}+1},$$

(12)

for any $\breve{e}\ge 1$. Using (12) and the monotonicity of $\mathcal{J}$, we obtain

$$\begin{array}{cc}\hfill \vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})& =\vartheta \left(\right(1-{\alpha }_{\breve{e}}){\hslash }_{\breve{e}}\oplus {\alpha }_{\breve{e}}{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},\mathsf{p})\le (1-{\alpha }_{\breve{e}})\vartheta ({\hslash }_{\breve{e}},\mathsf{p})+{\alpha }_{\breve{e}}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},\mathsf{p})\hfill \\ \hfill & =(1-{\alpha }_{\breve{e}})\vartheta ({\hslash }_{\breve{e}},\mathsf{p})+{\alpha }_{\breve{e}}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\mathcal{J}}_{\mathring{\varrho }}\mathsf{p})\hfill \\ \hfill & \le (1-{\alpha }_{\breve{e}})\vartheta ({\hslash }_{\breve{e}},\mathsf{p})+{\alpha }_{\breve{e}}\vartheta ({\hslash }_{\breve{e}},\mathsf{p})\le \vartheta ({\hslash }_{\breve{e}},\mathsf{p}),\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill \vartheta ({\mathsf{\ell }}_{\breve{e}},\mathsf{p})& =\vartheta ((1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}}){\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\oplus {\mathfrak{Y}}_{\breve{\mathfrak{e}}}{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\le (1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}})\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},\mathsf{p})+{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\hfill \\ \hfill & =(1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}})\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},\mathsf{p})+{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}},{\mathcal{J}}_{\mathring{\varrho }}\mathsf{p})\hfill \\ \hfill & \le (1-{\mathfrak{Y}}_{\breve{\mathfrak{e}}})\vartheta ({\hslash }_{\breve{e}},\mathsf{p})+{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\le \vartheta ({\hslash }_{\breve{e}},\mathsf{p}).\hfill \end{array}$$

(14)

By using (14), we get

$$\vartheta ({\hslash }_{\breve{e}+1},\mathsf{p})=\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}},\mathsf{p})\le \vartheta ({\mathsf{\ell }}_{\breve{e}},\mathsf{p})\le \vartheta ({\hslash }_{\breve{e}},\mathsf{p}).$$

(15)

Therefore, the inequality $\vartheta ({\hslash }_{\breve{e}+1},\mathsf{p})\le \vartheta ({\hslash }_{\breve{e}},\mathsf{p})$ is valid for all $\breve{e}\ge 1$. Consequently, the sequence $\left\{\vartheta \right({\hslash }_{\breve{e}},\mathsf{p}\left)\right\}$ is a decreasing sequence of real numbers that is bounded below by zero, ensuring its convergence. Consequently, the limit of $\vartheta ({\hslash }_{\breve{e}},\mathsf{p})$ as $\breve{e}$ approaches infinity exists.

Subsequently, we establish (ii). Let’s consider:

$$\underset{\breve{e}\to \infty }{lim}\vartheta ({\hslash }_{\breve{e}},\mathsf{p})=c.$$

(16)

Then, from (13) and (14), we have

$$\underset{\breve{e}\to \infty }{lim}\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\le c$$

(17)

and

$$\underset{\breve{e}\to \infty }{lim}\vartheta ({\mathsf{\ell }}_{\breve{e}},\mathsf{p})\le c.$$

(18)

Furthermore, utilizing (12) and the monotonicity property of $\mathcal{J}$, we obtain:

$$\underset{\breve{e}\to \infty }{lim\; sup}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\le \underset{\breve{e}\to \infty }{lim\; sup}\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\le c,$$

(19)

$$\underset{\breve{e}\to \infty }{lim\; sup}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}},\mathsf{p})\le \underset{\breve{e}\to \infty }{lim\; sup}\vartheta ({\mathsf{\ell }}_{\breve{e}},\mathsf{p})\le c,$$

(20)

and

$$\underset{\breve{e}\to \infty }{lim\; sup}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},\mathsf{p})\le \underset{\breve{e}\to \infty }{lim\; sup}\vartheta ({\hslash }_{\breve{e}},\mathsf{p})\le c.$$

(21)

Now, we have

$$c=\underset{\breve{e}\to \infty }{lim}\vartheta ({\hslash }_{\breve{e}+1},\mathsf{p})=\underset{\breve{e}\to \infty }{lim}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\mathsf{\ell }}_{\breve{e}},\mathsf{p}).$$

(22)

Consider

$$\vartheta ({\hslash }_{\breve{e}+1},\mathsf{p})\le (1-{\beta }_{\breve{e}})\vartheta ({\hslash }_{\breve{e}},\mathsf{p})+{\mathfrak{Y}}_{\breve{\mathfrak{e}}}\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p}).$$

We have

$${\mathfrak{Y}}_{\breve{\mathfrak{e}}}\left(\vartheta ({\hslash }_{\breve{e}},\mathsf{p})-\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})\right)\le \vartheta ({\hslash }_{\breve{e}},\mathsf{p})-\vartheta ({\hslash }_{\breve{e}+1},\mathsf{p}).$$

By using $0<\tilde{o}<{\beta }_{\breve{e}}<1,$ for all $\breve{e},$ we obtain

$$c\le \underset{\breve{e}\to \infty }{lim\; inf}\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p}),$$

(23)

which by using (17), yields

$$\underset{\breve{e}\to \infty }{lim}\vartheta ({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{p})=c.$$

(24)

Now, from (16), (21), (24) and Lemma 4, we obtain

$$\underset{\breve{e}\to \infty }{lim}\vartheta ({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}})=0.$$

□

Lemma 8.

Consider $\mathfrak{U}\ne \varnothing$ as a closed CS of a complete ordered CAT(0) space $(\mathcal{S},\prec )$, where $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ is a monotone $\mathring{\varrho }$-ENEM. Let ${\hslash }_1\in \mathfrak{U}$ be fixed st ${\hslash }_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1$. If the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ is defined by (9), then under the conditions Δ-${lim}_{\breve{e}}{\hslash }_{\breve{e}}=\hslash$ and ${lim}_{\breve{e}\to \infty }\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)=0$, it follows that ℏ serves as a FP of $\mathcal{J}$.

Proof. 

Given that $\Delta$-${lim}_{\breve{e}}{\hslash }_{\breve{e}}=\hslash$, applying Lemma 6 implies that ${\hslash }_{\breve{e}}⪯\hslash$ holds for all $\breve{e}\ge 1$. Subsequently, due to the non-expansiveness of $\mathcal{J}$ and the convergence ${lim}_{\breve{e}\to \infty }\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)=0$, it can be concluded that:

$$\begin{array}{cc}\hfill \vartheta \left({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\hslash }_{\breve{e}}\right)& \le \vartheta \left({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\right)+\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)\hfill \\ \hfill \underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\hslash }_{\breve{e}}\right)& \le \underset{\breve{e}\to \infty }{lim\; sup}\left[\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\right)+\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)\right]\hfill \\ \hfill & =\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\right)\le \underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left(\hslash ,{\hslash }_{\breve{e}}\right)=r\left(\hslash ,{\hslash }_{\breve{e}}\right).\hfill \end{array}$$

Hence, the uniqueness of AC implies ${\mathcal{J}}_{\mathring{\varrho }}\hslash =\hslash$, thus confirming the desired result. □

Theorem 1.

Let $\mathfrak{U}\ne \varnothing$ be a closed and CS of a complete ordered CAT(0) space $(\mathcal{S},⪯)$ and $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ be a monotone $\mathring{\varrho }$-ENEM and $Fix\left(\mathcal{J}\right)\ne \varnothing$. Fix ${\hslash }_1\in \mathfrak{U}$ st ${\hslash }_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1$. If $\left\{{\hslash }_{\breve{e}}\right\}$ is given by (9), then $\left\{{\hslash }_{\breve{e}}\right\}$Δ-converges to a FP of $\mathcal{J}$.

Proof. 

From Lemma 7, we have ${lim}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},\mathsf{p}\right)$ exists ∀$\mathsf{p}\in Fi\hslash \left(\mathcal{J}\right)$ so, the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ is bounded and ${lim}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}}\right)=0$.

Let $W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)=:\cup A\left(\left\{u_{\breve{e}}\right\}\right)$, where union is taken over all subsequences $\left\{u_{\breve{e}}\right\}$ of $\left\{{\hslash }_{\breve{e}}\right\}$. To demonstrate the $\Delta$-convergence of the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ to a FP of $\mathcal{J}$, we initially establish $W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)\subset Fix\left({\mathcal{J}}_{\mathring{\varrho }}\right)$, and subsequently show that $W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$ consists of a single element. To establish $W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)\subset Fix\left({\mathcal{J}}_{\mathring{\varrho }}\right)$, suppose $\mathsf{\ell }\in W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$. Hence, there exists a subsequence $\left\{{\mathsf{\ell }}_{\breve{e}}\right\}$ of $\left\{{\hslash }_{\breve{e}}\right\}$ st $A\left(\left\{{\mathsf{\ell }}_{\breve{e}}\right\}\right)=\mathsf{\ell }$. According to Lemma 3, there exists another subsequence $\left\{{\mathfrak{q}}_{\breve{\mathfrak{e}}}\right\}$ of $\left\{{\mathsf{\ell }}_{\breve{e}}\right\}$ st $\Delta$-${lim}_{\breve{e}}{\mathfrak{q}}_{\breve{\mathfrak{e}}}=\mathfrak{q}$ and $\mathfrak{q}\in \mathfrak{U}$. Given that ${lim}_{\breve{e}\to \infty }\vartheta \left({\mathcal{J}}_{\mathring{\varrho }}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)=0$ and $\left\{{\mathfrak{q}}_{\breve{\mathfrak{e}}}\right\}$ is a subsequence of $\left\{{\hslash }_{\breve{e}}\right\}$, it follows that ${lim}_{\breve{e}\to \infty }\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},{\mathcal{J}}_{\mathring{\varrho }}{\mathfrak{q}}_{\breve{\mathfrak{e}}}\right)=0$. By virtue of Lemma 8, we conclude that $\mathfrak{q}={\mathcal{J}}_{\mathring{\varrho }}\mathfrak{q}$ and thus $\mathfrak{q}\in Fix\left(\mathcal{J}\right)$.

Our objective now is to establish that $\mathfrak{q}=\mathsf{\ell }$. If it were the case that $\mathfrak{q}\ne \mathsf{\ell }$, then we would have:

$$\begin{array}{cc}\hfill \underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathfrak{q}\right)& <\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathsf{\ell }\right)\le \underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathsf{\ell }}_{\breve{e}},\mathsf{\ell }\right)\hfill \\ \hfill & <\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathsf{\ell }}_{\breve{e}},\mathfrak{q}\right)=\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\hslash }_{\breve{e}},\mathfrak{q}\right)=\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathfrak{q}\right)\hfill \end{array}$$

This leads to a contradiction since $\left\{{\mathfrak{q}}_{\breve{\mathfrak{e}}}\right\}$ satisfies the Opial condition, implying that $\mathfrak{q}=\mathsf{\ell }\in Fi\hslash \left(\mathcal{J}\right)$. The next step is to demonstrate that $W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$ consists of a single element only. To do this, consider a subsequence $\left\{{\mathsf{\ell }}_{\breve{e}}\right\}$ of $\left\{{\hslash }_{\breve{e}}\right\}$. Utilizing Lemma 3 once more, we can identify a subsequence $\left\{{\mathfrak{q}}_{\breve{\mathfrak{e}}}\right\}$ of $\left\{{\mathsf{\ell }}_{\breve{e}}\right\}$ st $\Delta$-${lim}_{\breve{e}}{\mathfrak{q}}_{\breve{\mathfrak{e}}}=\mathfrak{q}$. Let $A\left(\left\{{\mathsf{\ell }}_{\breve{e}}\right\}\right)=\mathsf{\ell }$ and $A\left(\left\{{\hslash }_{\breve{e}}\right\}\right)=\hslash$. We have previously established that $\mathsf{\ell }=\mathfrak{q}$. Therefore, it suffices to demonstrate that $\mathfrak{q}=\hslash$. In the scenario where $\mathfrak{q}\ne \hslash$, given that $\mathfrak{q}\in Fi\hslash \left(\mathcal{J}\right)$ and by utilizing Lemma 7, we find that $\left\{\vartheta \left({\hslash }_{\breve{e}},\mathfrak{q}\right)\right\}$ is convergent. Applying the uniqueness of the asymptotic center, we conclude that:

$$\begin{array}{cc}\hfill \underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathfrak{q}\right)& <\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},\hslash \right)\le \underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\hslash }_{\breve{e}},\hslash \right)\hfill \\ \hfill & <\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\hslash }_{\breve{e}},\mathfrak{q}\right)=\underset{\breve{e}\to \infty }{lim\; sup}\vartheta \left({\mathfrak{q}}_{\breve{\mathfrak{e}}},\mathfrak{q}\right),\hfill \end{array}$$

This conflicts with the claim that $\left\{{\mathfrak{q}}_{\breve{\mathfrak{e}}}\right\}$ satisfies the Opial condition. Thus, it follows that $\mathfrak{q}=\hslash$, establishing that $W_{\omega }\left(\left\{{\hslash }_{\breve{e}}\right\}\right)$ is a set containing only one element, which happens to be the FP of $\mathcal{J}$. As a result, the conclusion can be drawn. □

Theorem 2.

Consider a complete ordered CAT $\left(0\right)$ space $\mathcal{S}$ equipped with the partial ordering ‘⪯’, and let $\mathfrak{U}\ne \varnothing$ be a closed CS of $\mathcal{S}$. Suppose $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ is a monotone $\mathring{\varrho }$-ENEM with a nonempty set of fixed points $Fix\left(\mathcal{J}\right)$. Choose ${\hslash }_1\in \mathfrak{U}$ st ${\hslash }_1⪯{\mathcal{J}}_{\mathring{\varrho }}{\hslash }_1$. If the sequence $\left\{{\hslash }_{\breve{e}}\right\}$, defined as in Equation (9), converges to a FP of $\mathcal{J}$, then it is equivalent to ${lim\; inf}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},Fix({\mathcal{J}}_{)}\right)=0$.

Proof. 

When the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ approaches a point ℏ belonging to $Fix\left(\mathcal{J}\right)$, it can be observed that ${lim\; inf}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)=0$.

For the converse aspect, suppose that ${lim\; inf}_{\breve{e}\to \infty }\left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)=0$. By Lemma 7$\left(\mathrm{i}\right)$, we can derive that

$$\vartheta \left({\hslash }_{\breve{e}+1},\mathsf{p}\right)\le \vartheta \left({\hslash }_{\breve{e}},\mathsf{p}\right)\mathrm{for}\mathrm{any}\mathsf{p}\in Fi\hslash \left(\mathcal{J}\right),$$

thus, we obtain

$$\vartheta \left({\hslash }_{\breve{e}+1},Fix\left(\mathcal{J}\right)\right)\le \vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right).$$

Therefore, the sequence $\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)$ is decreasing and bounded below by zero. Consequently, we can deduce that the limit of $\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)$ exists as $\breve{e}$ approaches infinity. As

${lim\; inf}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)=0$, we can conclude that ${lim}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)=0.$

Now, we will demonstrate that the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ is a CS in $\mathfrak{U}$. Take any arbitrary $ϵ>0$. As ${lim\; inf}_{\breve{e}\to \infty }\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)=0$, there exists an integer ${\breve{e}}_0$ st ∀$\breve{e}\ge {\breve{e}}_0$, we obtain:

$$\vartheta \left({\hslash }_{\breve{e}},Fix\left(\mathcal{J}\right)\right)<{\displaystyle \frac{ϵ}{4}}.$$

Specifically.

$$inf\left\{\vartheta \left({\hslash }_{\breve{e}},\mathsf{p}\right):\mathsf{p}\in Fix\left(\mathcal{J}\right)\right\}<{\displaystyle \frac{ϵ}{4}},$$

so there must exist $\mathsf{p}\in Fix\left(\mathcal{J}\right)$ st

$$\vartheta \left({\hslash }_{{\breve{e}}_0},\mathsf{p}\right)<{\displaystyle \frac{ϵ}{2}}.$$

Thus, for $\breve{c},\breve{e}\ge {\breve{e}}_0$, we have

$$\left({\hslash }_{\breve{e}+\breve{c}},{\hslash }_{\breve{e}}\right)\le \vartheta \left({\hslash }_{\breve{e}+\breve{c}},\mathsf{p}\right)+\vartheta \left({\hslash }_{\breve{e}},\mathsf{p}\right)<2\vartheta \left({\hslash }_{{\breve{e}}_0},\mathsf{p}\right)<2{\displaystyle \frac{ϵ}{2}}=ϵ.$$

this demonstrates that the sequence $\left\{{\hslash }_{\breve{e}}\right\}$ is a CS. Since $\mathfrak{U}$ is a closed subset of a complete metric space $\mathcal{S}$, then $\mathfrak{U}$ itself is a CMS, and therefore, $\left\{{\hslash }_{\breve{e}}\right\}$ must converges in $\mathfrak{U}$. Let ${lim\; inf}_{\breve{e}\to \infty }{\hslash }_{\breve{e}}=q$.

Now, $\mathcal{J}$ is a MNEM and from Lemma 7(ii), we have ${lim}_{\breve{e}\to \infty }\vartheta \left(\mathcal{J}{\hslash }_{\breve{e}},{\hslash }_{\breve{e}}\right)=0$. Furthermore, based on the argument presented in Lemma $3.1$ of [45], it is straightforward to infer that ${\hslash }_{\breve{e}}⪯q$ for all $\breve{e}\ge 1$. Consequently, we obtain the following:

$$\begin{array}{cc}\hfill \vartheta (q,\mathcal{J}q)& \le \vartheta \left(q,{\hslash }_{\breve{e}}\right)+\vartheta \left({\hslash }_{\breve{e}},\mathcal{J}{\hslash }_{\breve{e}}\right)+\vartheta \left(\mathcal{J}{\hslash }_{\breve{e}},\mathcal{J}q\right)\hfill \\ \hfill & \le \vartheta \left(q,{\hslash }_{\breve{e}}\right)+\vartheta \left({\hslash }_{\breve{e}},\mathcal{J}\left({\hslash }_{\breve{e}}\right)\right)+\vartheta \left({\hslash }_{\breve{e}},q\right)\to 0\mathrm{as}\breve{e}\to \infty ,\hfill \end{array}$$

therefore, we have $q=\mathcal{J}q$, which implies that q belongs to $Fix\left(\mathcal{J}\right)$ □

5. Numerical Example

We begin this section by demonstrating an example of a $2/5$-MENEM, which is not a MNM. Then, we will show the convergence of IS to the FP.

Example 3.

Let $\mathcal{S}=[0,+\infty )$ be a CMS with the metric

$$\vartheta (\hslash ,\mathsf{\ell })=|\hslash -\mathsf{\ell }|,\hslash ,\mathsf{\ell }\in \mathcal{S},$$

and $\mathfrak{U}=[\frac{8}{10},5]$. We now define the order relation ⪯ by stating that $\hslash ⪯\mathsf{\ell }$ if and only if $\hslash \le \mathsf{\ell }$:

$$\left\{\begin{array}{cc}\hslash \le \mathsf{\ell }\hfill & \mathit{if}\hslash ,\mathsf{\ell }\in \left[\frac{8}{10},1.1\right]\mathit{or}\hfill \\ \hslash \le \mathsf{\ell }\hfill & \mathit{if}\hslash ,\mathsf{\ell }\in \left(1.1,5\right].\hfill \end{array}\right.$$

Let $\mathcal{J}:\mathfrak{U}\to \mathfrak{U}$ be defined by

$$\mathcal{J}(\hslash )=\left\{\begin{array}{cc}\frac{1}{\hslash }\hfill & \hslash \in \left[\frac{8}{10},1.1\right]\hfill \\ 2.9\hfill & \hslash \in \left(1.1,5\right].\hfill \end{array}\right.$$

st for $\mathring{\varrho }=2/5$, $\mathcal{J}$ is $2/5$-MENEM for all $\hslash \in \mathfrak{U}$ as follows:

$${\mathcal{J}}_{\mathring{\varrho }}(\hslash )=\left\{\begin{array}{cc}\frac{2+3{\hslash }^2}{5\hslash }\hfill & \hslash \in \left[\frac{8}{10},1.1\right]\hfill \\ 0.6\hslash +1.16\hfill & \hslash \in \left(1.1,5\right].\hfill \end{array}\right.$$

It is evident that $\mathcal{J}$ is not continuous at $\hslash =2$, indicating that The mapping is not nonexpansive, and when $\hslash =1$, it acts as the FP of $\mathcal{J}$. Furthermore, It is easily verified that, in accordance with the given ordering, $\mathcal{J}$ functions as a $2/5$-MENEM in the following manner:

• If $\hslash ,\mathsf{\ell }\in \left[\frac{8}{10},2\right]$ and $\hslash \le \mathsf{\ell }$.
  Since $\hslash \le \mathsf{\ell }$ and notice that $\mathcal{J}$ is increasing, we have $\frac{2+3{\hslash }^2}{5\hslash }\le \frac{2+3{\mathsf{\ell }}^2}{5\mathsf{\ell }}$ which gives

  $${\mathcal{J}}_{\mathring{\varrho }}\hslash \le {\mathcal{J}}_{\mathring{\varrho }}\mathsf{\ell }.$$
  So, $\mathcal{J}$ is a $2/5$-enriched monotone map. Now, consider

  $$\vartheta ({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\mathcal{J}}_{\mathring{\varrho }}\mathsf{\ell })=\vartheta ({\displaystyle \frac{2+3{\hslash }^2}{5\hslash }},{\displaystyle \frac{2+3{\mathsf{\ell }}^2}{5\mathsf{\ell }}})=\left|{\displaystyle \frac{2+3{\hslash }^2}{5\hslash }}-{\displaystyle \frac{2+3{\mathsf{\ell }}^2}{5\mathsf{\ell }}}\right|\le |\hslash -\mathsf{\ell }|=\vartheta (\hslash ,\mathsf{\ell }).$$
• If $:\hslash ,\mathsf{\ell }\in \left(2,5\right]$ and $\hslash \le \mathsf{\ell }$.
  Since $\hslash \le \mathsf{\ell }$, we have $0.6\hslash +1.16\le 0.6\mathsf{\ell }+1.16$ which gives

  $${\mathcal{J}}_{\mathring{\varrho }}\hslash \le {\mathcal{J}}_{\mathring{\varrho }}\mathsf{\ell }.$$
  So, $\mathcal{J}$ is a $2/5$-enriched monotone map. Now, consider

  $$\begin{array}{cc}\hfill \vartheta ({\mathcal{J}}_{\mathring{\varrho }}\hslash ,{\mathcal{J}}_{\mathring{\varrho }}\mathsf{\ell })& =\vartheta (0.6\hslash +1.16,0.6\mathsf{\ell }+1.16)=\left|0.6\hslash +1.16-0.6\mathsf{\ell }+1.16\right|\hfill \\ \hfill & =0.6|\hslash -\mathsf{\ell }|\le |\hslash -\mathsf{\ell }|=\vartheta (\hslash ,\mathsf{\ell }).\hfill \end{array}$$
  Thus, $\mathcal{J}$ is a $2/5$-MENEM.

From the following

Table 1. Comparison between MRN and New Iterative Sequence.

$\mathring{\mathsf{\varrho }}=2/5$		$\mathring{\mathsf{\varrho }}=1/2$		$\mathring{\mathsf{\varrho }}=3/4$		$\mathring{\mathsf{\varrho }}=4/5$		$\mathring{\mathsf{\varrho }}=99/100$
New	MRN	New	MRN	New	MRN	New	MRN	New	MRN
0.80000	0.80000	0.80000	0.80000	0.80000	0.80000	0.80000	0.80000	0.80000	0.80000
0.99617	0.98068	1.00020	1.01190	0.94844	1.06925	0.92853	1.07743	0.82721	1.09973
0.99986	0.99721	1.00000	1.00000	0.98795	0.98173	0.97616	0.97644	0.85512	0.95792
0.99999	0.99957	1.00000	1.00000	0.99728	1.00497	0.99229	1.00725	0.88014	1.01660
1.00000	0.99999	1.00000	1.00000	0.99939	0.99878	0.99754	0.99799	0.90155	0.99362
1.00000	1.00000	1.00000	1.00000	0.99987	1.00039	0.99922	1.00065	0.91952	1.00242
1.00000	1.00000	1.00000	1.00000	0.99997	1.00032	0.99975	0.99989	0.93431	0.99914
1.00000	1.00000	1.00000	1.00000	0.99999	0.99999	0.99992	1.00000	0.94662	1.00031
1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	0.99998	1.00000	0.95661	0.99999
1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	0.99999	1.00000	0.96484	1.00000
1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	0.97152	1.00000
1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	0.97690	1.00000

and

Table 2. Comparison of the iteration (9) with different iteration processes.

Initial Point	S-Iteration	Thakur-Iteration	F-Iteration	Abbas-Iteration	(9)
$0.8$	62	62	18	17	5
$0.9$	58	58	16	16	4

, it is clear that newly defined IS not only converges to the FP of $\mathcal{J}$ but it also show that it has a better rate of convergence than predefined IS.

Analyzing the results in the table leads to the following conclusions:

• The rate at which the New IS (9) converges for the MENEM being considered is influenced by both the parameter $\mathring{\varrho }$ and the initial point ${\hslash }_1$.
• If the initial point is ${\hslash }_1=0.8$ and $\mathring{\varrho }>\frac{1}{2}$, the convergence of the new IS slows down as the parameter $\mathring{\varrho }$ approaches 1 (see Figure 1).
• For ${\hslash }_1=0.8$ (refer to Table 1), the fastest convergence of the new IS occurs when the parameter $\mathring{\varrho }$ is at $\frac{1}{2}$ (after one iteration, the exact FP value is obtained).
• For ${\hslash }_1=0.8$, MRN IS converges as slowly as the value of the parameter $\mathring{\varrho }$ approaches 1.
• MRN IS converges faster than the New IS for the value of the parameter $\mathring{\varrho }=\frac{99}{100}$ and ${\hslash }_1=0.8$ (see Figure 2).
• We conclude that the convergent behaviour of the New and MRN IS is similar in terms of the ip and the parameter $\mathring{\varrho }$. Nevertheless, for all scenarios analyzed with parameters $\mathring{\varrho }\le \frac{4}{5}$ and an initial value of ${\hslash }_1=0.8$, the MRN IS demonstrates a slow convergence rate.
• For ip ${\hslash }_1=0.8$, ${\hslash }_1=0.9$ and for the value of parameter $\mathring{\varrho }=\frac{2}{5}$ (see Table 2), New IS converges faster than S, Thakur, F and Abbas IS.
• After examining the data presented in Table 1 and Table 2, it is evident that utilizing the New IS (refer to Figure 3) would be a preferable approach for estimating the FP of certain MENEM.

Now, we present the following open problem:

For what values of the parameter $\mathring{\varrho }$, New IS has a better rate of convergence than MRN-IS for the class of MENEM?

6. Application to Integral Equations

In this section, we use our iteration scheme (9) to find the solution of following integral equation:

$$\hslash (t)={\displaystyle \frac{k(t)+{\int }_0^{1}G(t,v,\hslash (v\left)\right)dv-(1-\mathring{\varrho })\hslash (t)}{\mathring{\varrho }}},t\in [0,1],$$

(25)

where

(i)

$k\in L^2([0,1],\mathbb{R})$,

(ii)

$G:[0,1]\times [0,1]\times L^2([0,1],\mathbb{R})\to \mathbb{R}$ is a measurable and satisfies the condition

$$0\le \left|G\right(t,v,\hslash )-G(t,v,\mathsf{\ell }\left)\right|\le ∥\hslash -\mathsf{\ell }∥$$

for $t,v\in [0,1],\mathring{\varrho }\in (0,1)$ and $\hslash ,\mathsf{\ell }\in L^2([0,1],\mathbb{R})$ such that $\hslash \le \mathsf{\ell }.$

Recall that, for all $\hslash ,\mathsf{\ell }\in L^2([0,1],\mathbb{R}),$ we have

$$\hslash \le \mathsf{\ell }\iff \hslash \left(t\right)\le \mathsf{\ell }\left(t\right),\mathrm{for}\mathrm{almost}\mathrm{every}t\in [0,1].$$

Next, assume that there exist a nonnegative function $f(·,·)\in L^2([0,1]\times [0,1])$ and $M<0.5$ such that

$$\left|G\right(t,v,\hslash \left)\right|\le f(t,v)+M\left|u\right|$$

for $t,v\in [0,1]$ and $\hslash \in L^2([0,1],\mathbb{R}).$

Let

$$A=\{\mathsf{\ell }\in L^2([0,1],\mathbb{R}):{∥\mathsf{\ell }∥}_{L^2([0,1],\mathbb{R})}\le \rho \}$$

where $\rho$ is sufficiently large, that is, A is the closed ball of $L^2([0,1],\mathbb{R})$ centered at 0 with radius $\rho .$ Define the operator $\mathcal{J}:L^2([0,1],\mathbb{R})\to L^2([0,1],\mathbb{R})$ by

$$\mathcal{J}(\mathsf{\ell }(t))={\displaystyle \frac{k(t)+{\int }_0^{1}G(t,v,\hslash (v\left)\right)dv-(1-\mathring{\varrho })\hslash (t)}{\mathring{\varrho }}}.$$

(26)

Then ${\mathcal{J}}_{\mathring{\varrho }}\left(A\right)\subset A,$ and it is monotone $\mathring{\varrho }$-enriched nonexpansive mapping.

It is worth mentioning that every Hilbert space is a CAT(0) space, and so is $L^2([0,1],\mathbb{R}).$ Taking $\mathfrak{U}=L^2([0,1],\mathbb{R})$ and $\mathcal{J}$ as in (26) in Theorem 1, we get the following result.

Theorem 3.

Under the above assumptions, the sequence generated by (9) converges to the solution of integral Equation (25).

7. Conclusions and Future Works

In this study, we explored the existence and iterative approximation of fixed points for monotone enriched nonexpansive mappings within ordered CAT(0) spaces. By introducing a novel iteration scheme, we demonstrated its convergence properties compared to existing iterations in the literature. Our findings not only extend previous results on enriched nonexpansive mappings but also provide a more efficient approach for approximating fixed points in nonlinear spaces.

Theoretical analysis confirmed the $\Delta$-convergence and strong convergence of the proposed iteration under specific conditions, reinforcing its applicability in the broader framework of fixed point theory. Additionally, numerical experiments validated the effectiveness of our method, further substantiating its advantage over conventional iterative techniques. As an application, we use our iteration scheme (9) to find the solution of an integral equation.

Future research could focus on generalizing these results to other classes of ordered metric spaces and exploring potential applications in optimization problems and nonlinear analysis. The insights gained from this work contribute to the ongoing development of iterative methods in mathematical analysis and computational mathematics.